Democratic National Committee Chairman Tom Perez acknowledged Sunday that the party “fell short” in the 2016 presidential primary even as he disputed charges that the process was rigged in Hillary Clinton’s favor.

Donna Brazile, who served as interim chair from July 2016 to February, disclosed in excerpts released Thursday from her upcoming book that the Clinton campaign and DNC had entered into a joint fundraising agreement before Mrs. Clinton won the Democratic presidential nomination.

DNC Deputy Chairman Keith Ellison said Friday that Ms. Brazile’s account “cannot simply be dismissed,” but Mr. Perez insisted that Mrs. Clinton won the Democratic nod in a process run by the states, not the DNC.

“Where I think both … [Elizabeth] Warren and Keith Ellison and myself, where we agree, is we have to earn the trust of the voters,” Mr. Perez said on NBC’s “Meet the Press.” “And during the process of the Democratic primary, we fell short in that, undeniably. And I accepted that responsibility.”

He said the party has made reforms, including plans to announce the Democratic candidate debate schedule in 2020 “before we know who the candidates are,” adding that “the No. 1 goal has to be to be fair and transparent.”

A number of Democrats have accused the DNC of tilting the playing field in Mrs. Clinton’s favor and against Vermont Sen. Bernard Sanders, her chief primary opponent.

Asked by CNN on Thursday if she thought the process was “rigged,” Ms. Warren said she did, while former Sanders campaign manager Jeff Weaver accused the DNC of running a “laundering operation.”

“We were not offered veto power on staff at the DNC, I can tell you that,” Mr. Weaver told The Washington Post. “This was a laundering operation. They’d go to fundraisers, they’d get a $350,000 check from donors, which was supposed to be divvied up. Instead of disbursing that money, they’d turn around and run a small-dollar fundraising to generate small contributions that went to the Clinton campaign.”

Mr. Perez, who took over as DNC chairman in February, agreed that the party has to “do better.”

“We have to do better is what we have to do,” Mr. Perez said. “And that’s why I was very clear during our primary campaign, during the campaign for DNC chair that we have to make sure that everybody feels at the end of the process that everyone got a fair shake.”